# Integers

Integers, or whole numbers (like 1, 2, 3, .. 13, 14, .. etc.) have puzzled man since prehistoric times. For instance the fact that some numbers like 6 (=1+2+3) and 28 (=1+2+4+7+14) are the sum of their divisors has fascinated early Greek mathematicians. Until a few decades ago number theory was a relatively 'useless' area of mathematics, but today number theory is at the core of our cryptographic systems that secure transactions over the internet. Here I will illustrate some properties part of these concepts using Scalable Vector Graphics (SVG). SVG is a family of specifications of an open standard by the World Wide Web Consortium for describing two-dimensional static and dynamic graphics. Graphs on these pages are generated in flight by your own browser.

# Divisors

Divisibility is an interesting property of integers. A number 'N' has non-trivial divisors if there is another number 'M' so that M*X=N. Nontrivial means M is not equal to 1 or N itself (or -1 or -N ...).
The so called 'prime' numbers have -n-o- nontrivial divisors. If we exclude 1 from the prime numbers (which is the common but arbitrary definition) they are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... Lots of questions: like is there an end to primes? (No there is not..) Are there arbitrary large gaps in between subsequent primes? (Yes.) How many primes differing only 2 (like 29 and 31) are there? (Infinite.) Etc.
We start off plotting N on an X-axis and on the y-axis the divisors N, this displays a beautifull pattern. In this case we include the trivial divisors: N itself and 1. In the 'millimeter paper' the most left down square has coordinates x=1 and y=1. Prime numbers are displayed by red lines.

This pattern keeps repeating itself if we 'travel' to the right. This is beatifully illustrated at divisorplot.com.